---
title: "Geometric elasticity: how thin things bend, buckle and snap"
description: "How thin strips, discs and folded tubes bend, buckle and snap: eight chapters on Euler buckling, snap-through, curved growth and Kresling origami, with Labs."
canonical_url: "https://sandboxphysics.com/stories/geometric-elasticity/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/geometric-elasticity/"
last_modified: "2026-10-31"
---

[Geometric elasticity · A course with experiments](https://sandboxphysics.com/stories/geometric-elasticity/)

FROM A BUCKLED STRIP TO A FOLDED TUBE

# Geometric elasticity: how thin things bend, buckle and snap

A thin sheet can bend for almost nothing, but stretching it costs far more. That one fact explains why a clamped strip buckles into an arch, why the arch jumps when pushed, why a growing disc curls into a dome and why a folded paper tube can click into a second shape and stay there. Each chapter derives one of these and checks it in a Lab.

Sandbox Physics · 2026-10-30

A strip between two clamps · Chapters 1–4

## Bend, buckle, snap

Find out why thin things bend instead of stretching, buckle a clamped strip into an arch, push the arch until it jumps, then work out how fast the jump grows and how the strip rings afterwards.

[Start with a thin strip →](https://sandboxphysics.com/stories/geometric-elasticity/why-thin-sheets-bend-instead-of-stretch/)

Shape from geometry · Chapters 5–8

## Growing discs and folded tubes

Give a disc lengths that no flat sheet can hold and find the thickness where it gives up and curves, fold a Kresling tube that clicks into a second shape, then read the snapping-beam paper figure by figure.

[Start with a growing disc →](https://sandboxphysics.com/stories/geometric-elasticity/growth-that-no-flat-sheet-can-hold/)

## Eight chapters, with examples and experiments

1. [01 L1 Why a thin sheet bends instead of stretching Why does a sheet of paper bend at a touch but hardly stretch at all? Read and try →](https://sandboxphysics.com/stories/geometric-elasticity/why-thin-sheets-bend-instead-of-stretch/)
2. [02 L2 A clamped strip buckles into an arch at the Euler load How far must the ends of a clamped strip be pushed together before it buckles, and how high does the arch then rise? Read and try →](https://sandboxphysics.com/stories/geometric-elasticity/euler-buckling-of-a-clamped-strip/)
3. [03 L2 Pushing an arch until it snaps When a blade pushes a buckled strip slowly down, where does the strip jump, and why there? Read and try →](https://sandboxphysics.com/stories/geometric-elasticity/pushing-an-arch-until-it-snaps/)
4. [04 L3 How fast a snap happens Once the blade lets go, how long does the strip take to snap, and what sets that time? Read and try →](https://sandboxphysics.com/stories/geometric-elasticity/how-fast-a-snap-happens/)
5. [05 L2 Growth that no flat sheet can hold What does it mean for growth to give a sheet a curved geometry, and why can no flat sheet have it? Read and try →](https://sandboxphysics.com/stories/geometric-elasticity/growth-that-no-flat-sheet-can-hold/)
6. [06 L3 The thickness where a disc buckles A disc is asked for a curved geometry. Below what thickness does it leave the plane, and how does its energy split once it has? Read and try →](https://sandboxphysics.com/stories/geometric-elasticity/the-thickness-where-a-disc-buckles/)
7. [07 L3 Kresling folds and their stress-free shapes Why does a twisted origami tube stay folded with nothing latching it, and which designs spring back instead? Read and try →](https://sandboxphysics.com/stories/geometric-elasticity/kresling-folds-and-stress-free-states/)
8. [08 L4 Reading the snapping-beams paper What does each figure of Pandey et al. (2014) show, how was it obtained, and which of its numbers can a reader check? Read and try →](https://sandboxphysics.com/stories/geometric-elasticity/reading-the-snapping-beams-paper/)

## Keep the experiments beside the chapters

The three Labs were built for this course. Open one beside a chapter and set it to the numbers of the worked example.

[Press a clamped arch until it snaps ↗](https://sandboxphysics.com/labs/arch-snap-through/) [Grow a flat disc into a dome ↗](https://sandboxphysics.com/labs/non-euclidean-plate/) [Press a folded Kresling tube ↗](https://sandboxphysics.com/labs/kresling-origami/)

Other elasticity Labs on the site

Each one takes a single idea from Chapters 1 and 2 on its own: a column buckling under load, a beam bent to read its modulus, and bending energy setting the shape of a membrane.

[When does a slender column buckle? ↗](https://sandboxphysics.com/labs/column-buckling/) [Bend a beam and read its modulus ↗](https://sandboxphysics.com/labs/beam-bending/) [Bending energy shapes a membrane vesicle ↗](https://sandboxphysics.com/labs/vesicle-shapes/)